 11.2 (Integral geometric measure) 

Here, we show that the methods of REF  apply, in
fact, to a special measure space. We keep the same notations as in
REF  and we let \({I}^k_\infty \)  be the integral
geometric outer measure on \(\mathbb {R}^m\)  defined in [1]} or
[2]}. The measure space \((\mathbb {R}^m,{B}(\mathbb {R}^m),{I}^k_\infty )\) 
is not semi-finite (for the case \(1=k=m-1\) , see
[1]}). Thus, recalling REF , we introduce the
following:
\(\check{{I}}^k_\infty (A) = \sup \left\lbrace  {I}^k_\infty (A \cap B) : B \in {B}(\mathbb {R}^m), B \subseteq A \text{ and } {I}^k_\infty (B) < \infty \right\rbrace ,\) 
